Nearest just interval

An irrational interval or ratio of frequencies given by a real number r has an infinite list of nearest just intervals; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call best rational approximations. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but |4/3 - √2| = .08088 whereas |3/2 - √2| = 0.08479, which is larger.

Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest.

The semiconvergents of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely best relative approximation. Here it is required that |qr - p| is less than |nr - m| for any n < q.

Examples

Approximations for Ratios (of Pure Intervals)

The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701.955000865... cents):

Step\EDO	log(Tenney Height)	size in cents	"error" in cents
...	...	...	...
1 \ 1	0.0	1200.0	498.04
1 \ 2	1.0	600.00	-101.96
2 \ 3	2.585	800.00	98.045
3 \ 5	3.907	720.00	18.045
4 \ 7	4.807	685.7143	-16.2407
7 \ 12	6.392	700.00	-1.955
17 \ 29	8.945	703.4483	1.4933
24 \ 41	9.943	702.43902	0.48402
31 \ 53	10.682	701.88679	-0.06821

• for approximations of the harmonic seventh see 7_4

Approximation for Logarihmic Measures

The 600-cent interval sqrt(2) (6 steps of 12edo, "Tritone") approximates following ratios:

freq. ratio	log2(Tenney Height)	size in cents	"error" in cents
...	...	...	...
1 / 1	0.0	0.0	600.0
3 / 2	2.585	701.96	101.96
4 / 3	3.585	498.04	-101.96
7 / 5	5.129	582.51	-17.49
17 / 12	7.672	603.000	3.000
24 / 17		597.000	-3.000
99 / 70		600.0883	0.0883
140 / 99		599.9117	-0.0883
...	...	...	...

The 300-cent interval 2^(1/4) (3 steps of 12edo, "minor third") approximates following ratios:

freq. ratio	log(Tenney Height)	size in cents	"error" in cents
...	...	...	...
1 / 1	0.0	0.0	300.0
6 / 5	4.907	315.64	15.64
13 / 11	7.160	289.21	-10.79
19 / 16	8.248	297.51	-2.49
25 / 21	9.036	301.84	1.84
...	...	...	...